# Full text of "Analytical Mechanics"

## See other formats

```FIELDS OF FORCE AND -NEWTONIAN POTENTIAL
Let A3 Fig. 105, be the body, 0 the point where the axis pierc plane of the paper, C the center of mass, and D its distance from th Then at the displaced position the potential energy is
U = mgh                            (0
= mgD (1 — cos0).
Therefore the torque experienced by the body is
= —mgrDsin 6.
This result may be easily verified by consid-               -pIG
ering the moments of the forces which act
upon the body.   The forces which act upon the body are the react
the axis and the weight of the body.   The moment of the react
nil; therefore the resultant moment is entirely due to the weigh
equals
Q — —mg«d = — mg D sin 0,
which is the result obtained by the other method.   The negative i introduced to indicate the fact that the rotation is clockwise.
173.  New Condition of Equilibrium. — Equations (I), and (II) provide us with a new condition for the equilit of conservative systems.    It was shown in Chapters I] III that a system is in equilibrium when the resultant and the resultant torque vanish.
Therefore setting F and G equal to zero in equa (I) and (II) we obtain
ds
= 0,
dd
where the differentiation in the first equation is with re; to any direction and that in the second with respect 1 angle about any axis. But when equations (III) are i fied,' U has a stationary value, that is, the value of U is e a minimum, or a maximum, or a constant. Therefor```